BASIN SCALE TSUNAMI PROPAGATION MODELING USING BOUSSINESQ MODELS: PARALLEL IMPLEMENTATION IN SPHERICAL COORDINATES ABSTRACT

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1 BASIN SCALE TSUNAMI PROPAGATION MODELING USING BOUSSINESQ MODELS: PARALLEL IMPLEMENTATION IN SPHERICAL COORDINATES J. T. Kirby, N. Pophe, F. Shi, S. T. Grilli 3 ABSTRACT We derive weakly nonlinear, weakly dispersive model equaions for propagaion of surface graviy waves in a shallow, homogeneous ocean of variable deph on he surface of a roaing sphere. A numerical scheme is developed based on he saggered-grid finie difference formulaion of Shi e al (). The model is implemened using he domain decomposiion echnique in conjuncion wih he message passing inerface (MPI). The efficiency ess show a nearly linear speedup on a Linux cluser. Relaive imporance of frequency dispersion and Coriolis force is evaluaed in boh he scaling analysis and he numerical simulaion of an idealized case on a sphere. INTRODUCTION The convenional models in he global-scale sunami modeling are based on he shallow waer equaions and neglec frequency dispersion effecs in wave propagaion. Recen sudies on sunami modeling revealed ha such sunami models may no be saisfacory in predicing sunamis caused by nonseismic sources (Løvhol e al., 8). For seismic sunamis, he frequency dispersion effecs in he long disance propagaion of sunami frons may become significan. The numerical simulaions of he 4 Indian Ocean sunami by Glimsdal e al. (6) and Grue e al. (8) indicaed he undular bores may evolve in shallow waer, as he phenomenon evidenced in observaions (Shuo, 985). In he simulaion for he same sunami by Grilli e al. (7), he dispersive effecs were quanified by running he dispersive Boussinesq model FUNWAVE (Kirby e al., 998) and he NSWE solver. Differences of up o % in surface elevaions, beween Boussinesq and NSWE simulaions, were found in deeper waer. Kulikov (5) performed a wavele frequency analysis based on saellie alimery daa recorded in he Bay of Bengal in deep waer, and showed he imporance of dispersive effecs on wave evoluion. He concluded ha a long wave model including he dispersion mechanism should be used for his even. For global wave propagaion, sphericiy and Coriolis effecs migh play roles in simulaing sunami signals a far disan ide gauges. Dispersive Boussinesq models such as FUNWAVE are usually developed in Caresian coordinaes for modeling ocean wave ransformaion from inermediae waer dephs o he coas. Løvhol e al. (8) recenly repored a Boussinesq model including spherical coordinaes and he Coriolis effec. The effecs of he earh's roaion and he imporance of Coriolis forces on he far-field propagaion across he Alanic Ocean were quanified in a model applicaion o a poenial sunami from he La Palma Island. Alhough he Boussinesq models are believed o be more accurae ools han sandard sunami models (based on shallow waer heory) in predicing dispersive sunami waves, he efficiency of Boussinesq modeling of sunamis is a concern. As poined ou by Yoon (), Boussinesq models consume huge compuer resources due o he implici naure of he soluion echnique used o deal wih dispersion erms. Some simulaions may involve a wide range of effecs of ineres, from propagaion ou of he generaion region, hrough propagaion a ocean basin scale, o runup and inundaion a affeced shorelines (Grilli e al., 7). Approaches for improving model efficiency can be found in some pracical applicaions such as using esed grids, unsrucured or curvilinear grids (Shi e al., ) and he parallelizaion of compuaional codes (Sianggang and Lyne, 5). In his sudy, we derived weakly nonlinear, weakly dispersive model equaions for surface graviy waves on he surface of a roaing sphere. A numerical model based on he equaions was implemened using he domain decomposiion echnique in conjuncion wih he message passing inerface (MPI). A numerical scheme is developed following he saggered-grid finie difference formulaion of Shi e al. (). The efficiency of he model was examined in es cases wih differen number of compuer processors. The relaive imporance of frequency dispersion and Coriolis force was evaluaed in boh he heory and he numerical simulaion of an idealized case. Cener for Applied Coasal Research, Universiy of Delaware Newark, USA, kirby@udel.edu Dep. Mahemaics, Faculy of Science, Chulalongkorn Univ., Thailand 3 Deparmen of Ocean Engineering Universiy of Rhode Island, Narraganse, USA

2 MODEL EQUATIONS IN SPHERICAL POLAR COORDINATES We consider moion in a fluid column of variable sill waer deph h (, ) where coordinaes r,, on he surface of a sphere, denoe radial disance from he sphere cener, laiude, and longiude. We denoe he radius o he spherical res surface as r and define a local verical coordinae z according o z = r r. The dimensional Euler equaions for incompressible inviscid flow are given by (Pedlosky 979, secion 6.) w w z + + ( v cos ) + u = r r cos r cos du u + ( w v an ) + Ω( w cos v sin ) = p d r ρr cos dv + ( vw + u an ) + Ω sinu = p d r ρr () () (3) dw ( u + v ) Ω cosu = p z g d r ρ where u and v are posiive in he Easerly ( ) and Norherly ( ) direcions respecively, and where w denoes verical velociy. The oal derivaive operaor is given by (4) d() u v = () + () + () + w () d r cos r z (5) Boundary condiions consis of a dynamic condiion specifying pressure p s on he waer surface as p (,, ) = ; z = η (6) s ogeher wih kinemaic consrains on he velociy field a he surface and boom boundary. The kinemaic surface boundary condiion (KSBC) is given by Dη = w ; z = η D (7) and he boom boundary condiion (BBC) is given by where D( h ) = w ; z =h D D() u v = () + () + () D r cos r (8) (9) Noe ha (8) allows for an imposed moion of he boom o be incorporaed. In Boussinesq or shallow waer heory, i is ypical o replace he local coninuiy equaion () wih a dephinegraed conservaion equaion for horizonal volume flux. Inegraing () over deph and employing he kinemaic boundary condiions (7) and (8) gives he exac equaion

3 η η { h } { h } () r () r h h cos urdz cos vrdz η η = () This will be simplified below based on scaling argumens.. Scaling Based on he usual noions for shallow waer scaling, we inroduce he lengh scales h, a and λ o denoe a characerisic deph, a characerisic ampliude, and a characerisic horizonal lengh scale. Due o he close ie beween source widh and generaed wave cres, λ would normally be idenified as a lenghscale W represening horizonal exen of he sunami source region. Combinaions of hese scales wih each oher and wih r lead o a family of dimensionless parameers: ε = h / r denoing he relaive deph or hickness of he ocean layer; µ h = /, he usual parameer characerizing frequency dispersion in Boussinesq heory; and δ = a / h, he shallow waer nonlineariy parameer. The parameer ( ) 3 ε akes on values of O a maximum, and will hus always be aken o indicae vanishingly small effecs when i occurs in isolaion. Based on his family of parameers, we inroduce dimensionless variables ( h, z ) η ( hz, ) = ; η = () h a u = δc = δ gh o denoe a scale for horizonal velociies, and le We ake verical velociy, so ha w denoe a scale for ( u, v ) w ( uv, ) = ; w= () u w Pressure is scaled by he weigh of he saic reference waer column, p p = (3) ρgh We firs inroduce a rescaling of he dimensionless laiude and longiude according o r µ (, ) = (, ) = (, ); λ ε (4) This gives horizonal coordinaes which will change by O () amouns over he wavelengh of a relaively shor wave. The nondimensional form of he coninuiy equaion () is hen given by (reaining erms o O( ε ) ) w µ ( εz) u cos [ wz + εw] + [ u + ( vcos ) ] = O( ε ) (5) indicaing ha w = µ u, as is usual in a Boussinesq model framework. Turning o he deph-inegraed mass conservaion equaion (), we inroduce he oal deph H = h+ δη (6) 3

4 and ge where δ H + ( Hu) + ( Hvcos ) = O( ε) cos δη ( uv, ) = ( uvdz, ) (8) h H (7) are deph averaged horizonal velociies, and where ime is scaled according o gh = ω = λ (9) Equaion (7) is he final volume flux conservaion equaion for he Boussinesq model sysem based on ( uv, ) developed below. In keeping wih he noion ha waves which are shor relaive o he basin scale ( λ / r or ε / µ ) may have frequencies which are high relaive o he earh's roaion rae Ω ( ω / Ω>> ), we inroduce he scaling Ω= = O( ε ). Turning o he easerly ( ) momenum equaion (), we obain ω µ ε u ε δ u f( v µ wan ) + δ u + vu + wuz uvan + p = O( ε) µ cos µ cos () where he Coriolis parameer f =Ω sin. The norherly ( ) momenum equaion (3) becomes ε u ε + + δ an + δ = ( ε) µ cos µ v fu v vv wvz u p O () The dimensionless z momenum equaion is given by u δµ w + δ w + vw + wwz fuco + ( p ) ( ) z + = O ε cos µ ε () In he following, we consider wo relaions beween ε and µ ; he regime µ O( ε ) /3 he shallow waer equaions, and he regime µ O( ε ). Shallow Waer Equaions =, which recovers =, which yields a Boussinesq approximaion. Mos heories of ransoceanic sunami propagaion are based on shallow waer equaions, in recogniion of he vanishing effec of dispersion ( µ ). In he presen discussion, his limi is obained in he limi of he horizonal lenghscale of wave moion approaching he horizonal scale of a global-sized ocean basin, or λ. This implies ha he raio ε / µ = O(), while erms proporional o µ appearing alone are r essenially he size of already negleced erms of O( ε ). In his combined limi, he verical momenum equaion (eq) reduces o he hydrosaic balance, and may be inegraed down from he free surface o yield ph = δη z (3) 4

5 where we subsequenly use p h o denoe he hydrosaic componen of pressure. This expression is used o evaluae pressure gradien erms in he horizonal momenum equaions, yielding he final se of shallow waer equaions H + ( Hu) ( Hvcos ) + = δ cos (4) u fv+ η = (5) cos v + fu+ η = (6) where in his limi he scaled laiude and longiude rever o he original values. Equaion (4) reains he possibiliy of describing wave generaion hrough a boom moion effec since h is scaled by h raher han wave ampliude..3 The Boussinesq Approximaion h which appears a O( / δ ) We now wish o reain dispersive effecs o leading order in he descripion of wave moion. Furher, in order o provide a uniformly valid model which can be used o describe nonlinear wave evoluion in shallow coasal margins as well as mainly linear evoluion in he deep ocean basin, we will reain he mechanics of he fully nonlinear Boussinesq model framework, following largely he approach of Wei e al (995) bu working in he framework of roaional flow. We will, for now, reain deph-averaged horizonal velociies ( uv, ) as dependen variables raher han velociy a a reference elevaion (as in Wei e al and subsequen effors). In developing he model, we seek o reain erms o conras o he classical Boussinesq approach, which would ake 4 O( δ, δµ, µ ) and higher. O ( µ ) wihou any runcaion in orders of δ. This is in = O( ) and runcae erms of δ µ In he derivaion, we will reain he effec of an imposed boom moion h (). The approximaion is /3 3 accompanied by he assumpion ha µ = O( ε ). (For ε = O( ), his implies a dispersion erm µ = O( ), which would be reasonable for he usual surface wave problem. This choice of scaling hen implies ha O( ε / µ ) = O( µ ), indicaing ha Coriolis erms and undiffereniaed advecive acceleraion erms are he same size as he leading order deviaion from hydrosaic of he pressure erm..3. Pressure and verical momenum Pressure in he sysem being considered will deviae from hydrosaic by nonhydrosaic componen by p%, we wrie O ( µ ) amouns. Denoing his p z = p z + p% z (,,, ) h(,,, ) δµ (,,, ) = δη z+ δµ p% (7) Inroducing (7 in () and inegraing up o he free surface (where p % = ) gives δη δη u p( z) = wdz δ w vw + wwz dz O( ε) z z cos % (8) The weakly nonlinear approximaion wih / O() δ µ = would reain 5

6 % (9) pz ( ) = wdz+ Oδ ( ) z.3. The verical srucure of velociies In order o use (8) o evaluae horizonal pressure gradiens, we need o esablish a relaion beween w and componens uv, hrough he coninuiy equaion (5), which simplifies o µ wz + u + ( vcos ) = O( ε) cos Inegraing (3) from h o z and using he boom boundary condiion gives z z ( h ) ( h ) w( z) = u dz v cos dz h cos cos δ = ( uh ( + z) ) ( vcos ( h+ z) ) h + O( µ ) cos cos δ (3) (3).3.3 Weakly nonlinear approximaion The sandard Boussinesq approximaion follows from he assumpion ha To he required order, he perurbaion o hydrosaic pressure is hen given by = O( ). δ µ z z p(,, z, ) = ( hu) ( hcos v) u ( vcos ) + BFT cos cos % (3) where BFT denoes forcing erms resuling from moion of he ocean boom, ( z δη) BFT = ( δη z) h + ( uh ) + ( v cos h ) δ cos (33) Using hese resuls, he horizonal momenum equaions are inegraed over deph and he perurbaion pressure is eliminaed using he previous resuls. The final dimensionless equaions are given by Figure : Relaive imporance of Coriolis force ( / ) ε µ and frequency dispersion ( ) µ = W. source widh h / µ wih varying inverse 6

7 { } H + ( Hu) + ( Hvcos ) = (34) δ cos u µ + δ + + η cos cos u fv u vu µ h h + u + ( vcos ) ( hu) + ( hcos v) cos 6 µ + = cos 4 ( BFT ) O( δ, δµ, µ ) (35) u + µ + δ + + η cos v fu v vv h + µ cos cos + µ ( BFT ) = O( δ, δµ, µ ) h { u ( vcos ) } {( hu) ( hcos v) } 4 (36).3.4 Dispersion versus Coriolis The orders of he frequency dispersions erms and Coriolis erms in Equaions (35) and (36) are O( ε / µ ), respecively. The relaive imporance of frequency dispersion and Coriolis force can be evaluaed using ε µ values in some specific cases. Figure illusraes variaions of µ and / O ( µ ) and µ and ε / µ respec o µ or h / W, where W represens widh of sunami source. Typically, for a source widh of km (4 Indian Ocean sunami), µ.5, he Coriolis effec is relaively much more imporan han dispersion as shown in Figure. For a narrow source wih a widh of 5 km, µ., he dispersive effec is as imporan as he Coriolis effec, and ges relaively more imporan as he source widh diminishes. 3 NUMERICAL APPROACH Dimensional forms of equaions (34) - (36) are discreized using he mehod described in Shi e al (). The sysem of equaions is rewrien in a compac form as H = E (37) U = F( η, uv, ) + F( v) + F( h, huv,, ) (38) where V = G( η, u, v) + G ( u ) + G ( h, h, u, v) (39) 7

8 E = ( Hu) ( Hvcos ) r cos + h h U = u+ u ( hu) r cos 6 F( η, u, v) = fv uu vu rcos r rcos h h ( ) = ( cos ) ( cos ) r cos 6 F v v h v h h h F ( h, h, u, v) = ( BFT ) ( h ) u rcos r cos + ( hu ) h h V = v+ ( vcos ) ( hcos v) r 6 cos cos g G( η, u, v) =fu uv vv η rcos r r h h ( ) = ( ) r 6 cos cos G u u hu h h G ( h, h, u, v) = ( BFT ) + ( hcos v) ( h cos v) r + cos cos g In Equaions (38) and (39), we inroduce FFF,,, GG, and G o represen separaely he erms wih differen properies. F and G include he Coriolis erms, pressure gradien erms and convecive erms; F and G are he linear dispersive erms; F and G are he erms associaed wih moion of he ocean boom. The arrangemen of cross-differeniaed and ime-derivaive erms on RHS of Equaions (U) and (V) makes he resuling se of LHS purely ridiagonal. A saggered grid in plane is employed. The firs-order spaial derivaive erms are discreized o fourh-order accuracy by using five-poin finie-differencing. The dispersive erms hemselves are finie-differenced only o second-order accuracy, leading o error erms of O(, ) relaive o he acual dispersive erms. In emporal discreizaion, he fourh-order Adams- Bashforh-Moulon predicor-correcor scheme is employed. The correcor sep is ieraed unil he error beween wo successive resuls reaches a required limi. 3. Parallelizaion In parallelizing he compuaional model, we use he domain decomposiion echnique o subdivide he problem ino muliple regions and assign each subdomain o a separae processor core. Each subdomain region conains an overlapping area of ghos cells wo rows deep, as dicaed by he 4h order compuaional sencil for he leading order non-dispersive erms. The Message Passing Inerface (MPI) wih non-blocking communicaion is used o exchange he daa in he overlapping region beween neighboring processors. Velociy componens are obained from Equaion (4) and (4) by solving ridiagonal marices using parallel pipelining ridiagonal solver described in Pophe e al (8). To invesigae performance of he parallel program, numerical simulaions of he idealized ocean case are esed wih differen number of processors (8, 6, 4, 3 and 36 processors) of a linux cluser locaed a Universiy of Delaware. The cluser has 8 compuing nodes wih 4 AMD Operon 87 dual cores processors. Each node has 3 GByes of main memory. The es case is se up in he compuaional domain of 9 o W - 9 o E and 6 o S - 6 o N discreized ino a numerical grid, and he iniial wave is generaed using an Okada source. η 8

9 Figure : Variaion in model performance wih number of processors for a domain. Sraigh line indicaes arihmeic speedup. Acual performance shown by green line. Figure shows he model speedup versus number of processors. I can be seen ha a super linear speedup is obained for he cases of 8 and 6 processors. This is due o he cache effec. When he size of sub-domain becomes small, he daa se accessed frequenly can fi ino caches and he memory access ime reduces dramaically. For he case wih 4, 3 and 36 processors, speedup decreases and drops below linear speedup. This is due o he communicaion cos. The overlapping area increases wih he number of processors. 4 AN OKADA SOURCE IN AN IDEALIZED OCEAN: SOURCE SIZE AND CORIOLIS EFFECTS To perform model ess wih he spherical Boussinesq model, we generae a model grid in spherical coordinaes and specify a fla boom bahymery in he ocean basin. The sunami source is based on he sandard half-plane soluion for an elasic dislocaion wih maximum slip (Okada, 985). Okada's soluion is implemened in ''Tsunami open and progressive iniial condiions sysem" (TOPICS) which provides he verical co-seismic displacemens as oupus. As an example, we specify a norh-souh oriened planar faul wih cenroid locaed a he equaor. The sunami source calculaed by TOPICS is ransferred and linearly superimposed ino he sperical Boussinesq model, as an iniial free surface condiion as shown in Figure (3). Figure 4 demonsraes he evoluion of sunami wave fron in ime. The wave dispersion effec can be recognized by a rain of waves near he wave fron in cases wih narrow sunami sources. The Coriolis effec can be indicaed by comparing model resuls wih and wihou Coriolis force. Figure 5 shows he relaive differences of surfave elevaion beween models wih and wihou Coriolis force. Figure 3: Source geomery for idealized ess of Coriolis and dispersion effecs. Boom displacemen resuling from an Okada source aligned in Norh-Souh direcion. 9

10 Figure 4: Evoluion of wave fron in ime in a consan deph ocean ( =,, 4, 6 hr). 5 CONCLUSIONS In his sudy, weakly nonlinear, weakly dispersive model equaions were derived for basin scale sunami propagaion on he surface of a roaing sphere. We inroduce he scaling parameers represening respecively hickness of ocean layer, frequency dispersion in Boussinesq heory and shallow waer nonlineariy for he Boussinesq approximaion. The derivaion followed he fully nonlinear Boussinesq model framework of We e al. (995) bu for roaional flow. The dispersive effecs were reained as he leading order in he descripion of wave moion. A numerical scheme is developed based on he saggered-grid finie difference formulaion of Shi e al (). The model is implemened using he domain decomposiion echnique in conjuncion wih he message passing inerface (MPI). The efficiency ess show a nearly linear speedup on a Linux cluser. Relaive imporance of frequency dispersion and Coriolis force is evaluaed in boh he scaling analysis and he numerical simulaion of an idealized case on a sphere. Figure 5: Relaive differences (%) of surface elevaion beween models wih and wihou Coriolis force. Acknowledgemens N. Pophe was suppored by he Deparmen of Civil and Environmenal Engineering, Universiy of Delaware, and by he Deparmen of Ocean Engineering, Universiy of Rhode Island. References Grue, J., Pelinovsky, E. N., Frucus, D., Talipova, T. and Kharif, C., 8, ``Formaion of undular bores and soliary waves in he Srai of Malacca caused by he 6 December 4 Indian Ocean sunami'', J. Geophys. Res., 3, C58, doi:.9/7jc4343.

11 Glimsdal, S., G. Pedersen, K. Aakan, C. B. Harbiz, H. Langangen, and F. Løvhol, 6, ``Propagaion of he Dec. 6, 4 Indian Ocean sunami: effecs of dispersion and source characerisics'', In. J. Fluid Mech. Res., 33(), Grilli, S. T., Ioualalen, M., Asavanan, J., Shi, F., Kirby, J. T. and Was, P., 7, ``Source consrains and model simulaion of he December 6, 4 Indian Ocean sunami'', J. Waerway, Por, Coas. and Ocean Engrng., 33, Imamura, F., N. Shuo, and C. Goo, 988, ``Numerical simulaion of he ransoceanic propagaion of sunamis'', presened a Sixh Congress of he Asian and Pacific Regional Division, In. Assoc. Hydraul. Res., Kyoo, Japan. Kirby, J. T., G. Wei, Q. Chen, A. B. Kennedy, and R. A. Dalrymple, 998, ``Fully nonlinear Boussinesq wave model documenaion and user's manual'', Research Rep. CACR-98-6, Cener for Appl. Coasal Res., Univ. of Delaware, Newark. Kulikov, E., 5, ``Dispersion of he Sumara sunami waves in he Indian Ocean deeced by saellie alimery.'' Rep. from P. P. Shirshov Insiue of Oceanology, Russian Academy of Sciences, Moscow. Løvhol, F., Pedersen, G. and Gisler, G., 8, ``Oceanic propagaion of a poenial sunami from he La Palma Island'', J. Geophys. Res., 3, C96, doi:.9/7jc463. Okada, Y., 985, Surface deformion due o shear and ensile fauls in a half-space, Bull. Seismol. Soc. Am., 75(4), Pedlosky, J., 979, Geophysical Fluid Dynamics, Springer-Verlag, New York, pp.64. Pophe, N., 8, ``Parallel compuaion for sunami'', M.S. Thesis, Chulalongkorn Universiy Pophe, N., Kaewbanjak, N., Asavanan, J. and Ioualalen, M., 8, ``Parallelizaion of fully nonlinear Boussinesq equaions for sunami simulaions: new approach on higher grid resoluion for sunami simulaion using parallelized fully nonlinear Boussinesq Equaions'', submied o Compuer and Fluids. Shi, F., Dalrymple, R. A., Kirby, J. T., Chen, Q. and Kennedy, A.,, ``A fully nonlinear Boussinesq model in generalized curvilinear coordinaes'', Coasal Engrng., 4, Shuo, N., 985, ``The Nihonkai-Chubu earhquake sunami on he norh Akia Coas", Coasal Eng. Jpn., 8, Sianggang, K. and Lyne, P., 5, ``Parallel compuaion of a highly nonlinear Boussinesq equaion model hrough domain decomposiion'', Inernaional Journal for Numerical Mehods in Fluids, 49, Tappin, D. R., Was, P. and Grilli, S. T., 8, ``The Papua New Guinea sunami of 7 July 998: anaomy of a caasrophic even'', Na. Hazards Earh Sys. Sci., 8, Wei, G., Kirby, J. T., Grilli, S. T. and Subramanya, R., 995, ``A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear, unseady waves'', J. Fluid Mech., 94, 7-9. Yoon, S. B.,, ``Propagaion of disan sunamis over slowly varying opography'', J. Geophys. Res., 7(C), 34, doi:.9/jc79.